Will it converge?

Calculus Level 5

Let a series be defined, for some real value of \(a_0\) and \(\alpha(>0)\) as:

\[\displaystyle \large a_n=a_{n-1}+\alpha \sin(a_{n-1})\]

\[X=\lim _{ n\rightarrow \infty }{ { a }_{ n } } \]

Let \(S\) be the sum of all the distinct values of \(X\) as \(a_0\) varies from \([1,100]\) and \(\alpha\) varies over all positive reals.

Find \[\left\lfloor 10^4S \right\rfloor \]

Details and Assumptions:

  1. \(X\) must be finite.

  2. \(\alpha\) is a variable independent of \(a_0\) and can take any positive value.

This is part of my set Powers of the ordinary.

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